Scalar conservation laws with stochastic discontinuous flux function

TL;DR

This paper develops a framework for stochastic conservation laws with discontinuous flux functions, proving well-posedness, and introduces a novel meshing strategy for numerical approximation that reduces variance.

Contribution

It introduces an admissibility criterion for stochastic discontinuous-flux laws, proves their well-posedness, and develops a new meshing method to improve numerical solutions.

Findings

The new meshing strategy reduces sample-wise variance.

The framework ensures pathwise existence and uniqueness of solutions.

Numerical experiments demonstrate the effectiveness of the approach.

Abstract

A variety of real-world applications are modeled via hyperbolic conservation laws. To account for uncertainties or insufficient measurements, random coefficients may be incorporated. These random fields may depend discontinuously on the state space, e.g., to represent permeability in a heterogeneous or fractured medium. We introduce a suitable admissibility criterion for the resulting stochastic discontinuous-flux conservation law and prove its well-posedness. Therefore, we ensure the pathwise existence and uniqueness of the corresponding deterministic setting and present a novel proof for the measurability of the solution, since classical approaches fail in the discontinuous-flux case. As an example of the developed theory, we present a specific advection coefficient, which is modeled as a sum of a continuous random field and a pure jump field. This random field is employed in the stochastic conservation law, in particular a stochastic Burgers' equation, for numerical experiments. We approximate the solution to this problem via the Finite Volume method and introduce a new meshing strategy that accounts for the resulting standing wave profiles caused by the flux-discontinuities. The ability of this new meshing method to reduce the sample-wise variance is demonstrated in numerous numerical investigations.